2. Proposed Scheme and Simulation Parameters

We consider here an unusual source of THz radiation, namely, from a plasma cylinder with external radial electric field and axial magnetic field, as shown in Figure 1. The electric field producing the plasma could be from a radial capacitor discharge and the magnetic field from a solenoid coil outside the cylinder. The electron motion is governed by

(1) $\begin{array}{}{d}_t\left(\gamma u\right)=-e\left(E+\frac{1}{c}u\times B\right),\end{array}$

(2) $\begin{array}{}{d}_t\gamma =\frac{e}{m{c}^2}u\times E,\end{array}$

where u is the electron velocity, γ is the relativistic factor, and e, m, and c are the electron charge and mass, and the speed of light, respectively. The dynamics of both electrons and (hydrogen) ions are fully included in the PIC simulations [Reference Xu, Chang, Zhuo, Cao and Yue28]. However, as expected, within the picosecond time of interest here, the electron motion is dominant. The constant embedded electric and magnetic fields are $E_0=E_0\hat{r}$ and $B_0=B_0\hat{z}$ , where E ₀ = 3.4 × 10¹⁰ V/m and B ₀ = 11.4 T, which we note are readily realizable. The cylindrical plasma cylinder is of radius R = 3286 μm and initially of uniform density n ₀ = 1.26 × 10²⁰ m⁻³. The temperature of the electrons and ions is 300 eV. The parameters have been chosen such that the electron cyclotron frequency ${\omega }_c=e{B}_0/mc$ and plasma frequency ${\omega }_{pe}={\left(4\pi e^2{n}_0,/,m\right)}^{1/2}$ are both in the THz regime, so that resonance of the electron oscillations can be expected.

Figure 1: The plasma cylinder with external radial electric field E ₀ and axial magnetic field B ₀, where E ₀ = 3.4 × 10¹⁰ V/m and B ₀ = 11.4 T. The (hydrogen) plasma density is n ₀ = 1.26 × 10²⁰ m⁻³.

To investigate the evolution of such a plasma cylinder, we use the 2D3V (two-dimensional in space and three-dimensional in velocity) PIC simulation code LAPINE [Reference Xu, Chang, Zhuo, Cao and Yue28]. The size of the simulation box is 9418 μm × 9418 μm. The spatial mesh has 1024 × 1024 cells, containing 4 × 10⁷ each of electrons and ions, and the simulation time step is 13 fs. As mentioned, the ion dynamics is fully included, as they might play a role in the long-time behavior of the system.

3. Initial Stage of the Evolution

Figure 2 shows the self-consistently induced electric field ΔE_r at t = 13.18 ps and t = 16.11 ps. In this beginning stage of the evolution, besides the cyclotron motion, the plasma electrons are driven radially by external electric field E ₀ and acquire a radial velocity υ _r. The induced electric field ΔE_r, of order hundreds V/m, is much weaker than E ₀. The Lorentz force from B ₀ and υ _r also cause the electrons to move in the azimuthal direction, and the resulting azimuthal velocity in turn causes the electrons to move in the radial direction, leading to the oscillations. The much heavier ions have hardly moved (not shown).

Figure 2: Snapshots of the induced electric field ΔE_r at (a) t = 13.18 ps and (b) t = 16.11 ps. Note that the induced electric field is much weaker than the embedded electric field.

The evolution of the current density distribution in the early stage, involving about two oscillation periods, is shown in Figure 3. Figure 3(a) shows that, at t = 11.72 ps, the current density near the cylinder center is larger than that near the boundary. Furthermore, the local current vectors (black arrows) indicate that the stronger current is in the anticlockwise direction (note that the electrons near the center rotate in the clockwise direction). For t = 13.18 ps, Figure 3(b) shows that the outward-moving electrons approach the boundary of the circular column. They are then reflected by the sheath there and move inwards toward the center, as can be seen in Figure 3(c). Roughly, Figures 3(a) and 3(c) correspond to near-minimum amplitudes and Figures 3(b) and 3(d) to near-maximum amplitudes, of the current oscillations. The evolution of the current density in Figure 3 shows that besides rotating in the anticlockwise direction, the electrons oscillate in a rather complicated manner.

Figure 3: Snapshots of the current distribution at (a) t =11.72, (b)t =13.18, (c) t =14.65, and (d) t =16.11 ps. The black arrows represent local current vectors. Close inspection shows that the currents oscillate both radially and azimuthally. The white dashed circles mark the boundary of the initial plasma cylinder.

4. Long-Time Behavior

Figure 3 shows the electron dynamics in the initial two oscillation cycles. To see the behavior at longer times, we now focus on a fixed location inside the plasma cylinder. Figures 4(a) and 4(c) show the evolution of the radial and azimuthal currents at (x, y) = (7346.04 μm, 7346.04 μm). One can see that, during the 80 ps long interval, the electron currents J _r and J _φ have undergone many oscillation cycles. Figures 4(b) and 4(d) show the corresponding frequency spectra. We can see in both the radial and azimuthal spectra a dominant peak at f∼0.35 THz (or ω ∼ 2.2 rad/sec). In fact, their overall spectral profiles are almost identical, suggesting that a 2D normal-mode-like structure has formed.

Figure 4: Evolution of the (a) radial current density J _r and (c) azimuthal current density J _φ at (x, y) = (7346.04 μm, 7346.04 μm). (b, d) The corresponding frequency spectra.

Figure 5 shows the evolution of the total current density $\sum \left(J\left(x\right)\right)$ in the simulation box. Figure 5(a) shows that the initial oscillations of the total current density are of large amplitude (to be expected since the initial plasma and fields do not form a self-consistent system), and the corresponding rate of decrease of the envelope is − 4.9 A/m²/ps. At later times, the rate of decrease becomes − 0.0012 A/m²/ps. Figure 5(b) shows the corresponding frequency spectrum, showing that the characteristic frequency of the current oscillations at this location is also ∼0.35 THz.

Figure 5: Evolution of the (a) total current density $\sum \left(J\left(x\right)\right)$ and (b) its frequency spectrum.

5. Terahertz Radiation

Using the temporal and spatial current density from the simulation, one can obtain the far-field radiation. The total electromagnetic energy per unit solid angle Ω per unit frequency is given by the following equation [Reference Cao, Yu and Xu15, Reference Jackson29]:

(3) $\begin{array}{}\frac{{\text{d}}^{2}I}{\text{d}\omega \text{d}\Omega }\approx \frac{{\omega }^2}{4{\pi }^2{c}^3}{\left(\int \text{d}t\int d^{3}x\hat{n}\times \left(\hat{n}\times J\left(x,t\right)\right)e^{i\omega \left(t-\hat{n}\cdot x/c\right)}\right)}^2,\end{array}$

where $\hat{n}$ is the direction of the observer with respect to the radiating body. The intensity distribution of the radiation in the z-direction is then [Reference Cao, Yu and Xu15]

(4) $\begin{array}{}\frac{d^{2}I}{\text{d}\omega \text{d}\Omega }\approx \frac{{\omega }^2}{4{\pi }^2{c}^3}{\left(\int \text{d}tJ\left(t\right)e^{i\omega t}\right)}^2,\end{array}$

where $J\left(t\right)=\int d^{3}xJ\left(x,t\right)$ . The corresponding result is shown in Figure 6. We can see that the radiation spectrum has a broad peak centered at f ₀∼0.35 THz, as to be expected in view of the current density spectra discussed above. The full-width-at-half-maximum bandwidth Δf/f ₀ is about 0.057.

Figure 6: The frequency spectrum of the THz radiation. The peak is at ∼0.35 THz, and the bandwidth is ∼5.7%.

6. Summary

In this paper, we have shown that a bounded plasma cylinder with embedded radial electric and axial magnetic fields can emit T-rays if the parameters of the plasma and the electric and magnetic fields are such that the electron plasma and cyclotron frequencies nearly match and are in the THz regime. As a result, resonance-like behavior of the electrons can occur. The electrons not only rotate in the azimuthal direction but also oscillate in both the radial and azimuthal directions. The resulting currents can emit THz radiation. Finally, it may be of interest to note that, in a nonperturbative analysis of the cold fluid and Maxwell equations, where all the electric and magnetic fields are self-consistent, similar coupled oscillation behavior of the charged particles can exist in an unbounded and self-consistently expanding plasma [Reference Karimov, Stenflo and Yu30].